The logarithmic function you mentioned has a domain of $(-2, \infty)$, which means that the input to the logarithmic function (inside the logarithm) is greater than $-2$. If this logarithmic function has an inverse, the inverse would be an exponential function.

Step-by-step reasoning:

1. Domain of the logarithmic function:
   $(-2, \infty)$ means that the logarithmic function is defined for all values greater than $-2$.

2. Range of the logarithmic function:
   The range of a standard logarithmic function is $(-\infty, \infty)$, as logarithms can output any real number.

3. Inverse of a logarithmic function (exponential function):
   The inverse of a logarithmic function is an exponential function. The domain of the original logarithmic function becomes the range of the inverse exponential function, and the range of the logarithmic function becomes the domain of the exponential function.

4. Range of the exponential function (inverse):
   The range of the exponential function will match the domain of the logarithmic function, so the range of the exponential child function will be $(-2, \infty)$.

Thus, the range of the exponential child function is: $(-2, \infty)$

Would you like further details on how these functions interact, or any specific questions?

Here are 5 related questions to expand on this topic:

1. What is the general relationship between logarithmic and exponential functions?
2. How do you find the inverse of a logarithmic function?
3. Can the base of a logarithmic or exponential function affect its domain or range?
4. How would transformations (shifts, stretches) affect the domain and range of logarithmic or exponential functions?
5. How do logarithmic and exponential equations appear in real-world applications?

Tip: Always verify whether the logarithmic function includes a base shift or other transformations before finding its inverse, as they affect the domain and range.